= Repeated-root parity-check matrix
If the <generator polynomial of a cyclic code> has distinct roots $\alpha_i$ with multiplicities $m_i$, a word belongs to the code exactly when $c^{[\ell]}(\alpha_i)=0$ for $0\leq\ell<m_i$, where $c^{[\ell]}$ is its <Hasse derivative>. The corresponding rows have entries $\binom j\ell\alpha_i^{j-\ell}$ for $j\geq\ell$ and zero otherwise. Expansion in a <basis> of the <splitting field> gives equations over the original <finite field>; an independent selection has <rank> $\sum_i m_i$.
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